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[Axiom-developer] 20070812.01.tpd.patch applied to silver
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20070812.01.tpd.patch applied to silver |
|
Date: |
Mon, 27 Aug 2007 02:31:36 -0500 |
diff --git a/changelog b/changelog
index 0aeec85..19aff4b 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,4 @@
+20070812 tpd re-merge input branch
20070811 tpd src/input/Makefile add classtalk, calcprob
20070811 tpd src/input/limit.input.pamphlet updated with new tests
20070811 tpd src/input/intbypart.input.pamphlet updated with new tests
diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet
index 621d8ed..34850f2 100644
--- a/src/input/Makefile.pamphlet
+++ b/src/input/Makefile.pamphlet
@@ -292,9 +292,10 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress
alist.regress \
binary.regress bop.regress bstree.regress bouquet.regress \
bug10069.regress \
bugs.regress bug10312.regress bug6357.regress bug9057.regress \
+ calcprob.regress \
calculus2.regress calculus.regress cardinal.regress card.regress \
carten.regress cclass.regress char.regress ch.regress \
- chtheorem.regress \
+ chtheorem.regress classtalk.regress \
clifford.regress clif.regress coercels.regress collect.regress \
complex.regress conformal.regress \
constant.regress contfrac.regress contfrc.regress \
@@ -486,10 +487,10 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input
${OUT}/algfacob.input \
${OUT}/bernpoly.input ${OUT}/binary.input ${OUT}/bop.input \
${OUT}/bouquet.input ${OUT}/bstree.input ${OUT}/bug6357.input \
${OUT}/bug9057.input ${OUT}/bug10069.input ${OUT}/bug10312.input \
- ${OUT}/calculus.input \
+ ${OUT}/calcprob.input ${OUT}/calculus.input \
${OUT}/cardinal.input ${OUT}/card.input ${OUT}/carten.input \
${OUT}/cclass.input ${OUT}/cdraw.input ${OUT}/char.input \
- ${OUT}/ch.input ${OUT}/chtheorem.input \
+ ${OUT}/ch.input ${OUT}/chtheorem.input ${OUT}/classtalk.input \
${OUT}/clifford.input ${OUT}/clif.input \
${OUT}/coercels.input ${OUT}/collect.input ${OUT}/color.input \
${OUT}/complex.input ${OUT}/cone.input ${OUT}/conformal.input \
@@ -669,12 +670,13 @@ DOCFILES= \
${DOC}/c06fqf.input.dvi ${DOC}/c06frf.input.dvi \
${DOC}/c06fuf.input.dvi ${DOC}/c06gbf.input.dvi \
${DOC}/c06gcf.input.dvi ${DOC}/c06gqf.input.dvi \
- ${DOC}/c06gsf.input.dvi ${DOC}/calculus2.input.dvi \
+ ${DOC}/c06gsf.input.dvi ${DOC}/calcprob.input.dvi \
+ ${DOC}/calculus2.input.dvi \
${DOC}/calculus.input.dvi ${DOC}/cardinal.input.dvi \
${DOC}/card.input.dvi ${DOC}/carten.input.dvi \
${DOC}/cclass.input.dvi ${DOC}/cdraw.input.dvi \
${DOC}/char.input.dvi ${DOC}/ch.input.dvi \
- ${DOC}/chtheorem.input.dvi \
+ ${DOC}/chtheorem.input.dvi ${DOC}/classtalk.input.dvi \
${DOC}/clifford.input.dvi ${DOC}/clif.input.dvi \
${DOC}/coercels.input.dvi ${DOC}/collect.input.dvi \
${DOC}/color.input.dvi ${DOC}/complex.input.dvi \
diff --git a/src/input/calcprob.input.pamphlet
b/src/input/calcprob.input.pamphlet
new file mode 100644
index 0000000..f41808d
--- /dev/null
+++ b/src/input/calcprob.input.pamphlet
@@ -0,0 +1,124 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input calcprob.input}
+\author{Timothy Daly}
+\maketitle
+\begin{abstract}
+Cover a range of calculus problems
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+<<*>>=
+)spool calcprob.output
+)set message test on
+)set message auto off
+)clear all
+
+--S 1
+solve(3*x-(x-7)=4*x-5,x)
+--R
+--R (1) [x= 6]
+--R Type: List Equation Fraction Polynomial
Integer
+--E 1
+
+--S 2
+solve(4*x-3*y=9,y)::List Equation Polynomial Fraction Integer
+--R
+--R 4
+--R (2) [y= - x - 3]
+--R 3
+--R Type: List Equation Polynomial Fraction
Integer
+--E 2
+
+--S 3
+solve(A*x+B*y=C,y)
+--R
+--R - A x + C
+--R (3) [y= ---------]
+--R B
+--R Type: List Equation Fraction Polynomial
Integer
+--E 3
+
+--S 4
+m:=3*x-4*(x-(2/3)*y)=(4/5)*x-(7*y+3)
+--R
+--R 8 4
+--R (4) - y - x= - 7y + - x - 3
+--R 3 5
+--R Type: Equation Polynomial Fraction
Integer
+--E 4
+
+--S 5
+n:=solve(m*15,y)
+--R
+--R 27x - 45
+--R (5) [y= --------]
+--R 145
+--R Type: List Equation Fraction Polynomial
Integer
+--E 5
+
+--S 6
+p:=n.1*145-27*x
+--R
+--R (6) 145y - 27x= - 45
+--R Type: Equation Fraction Polynomial
Integer
+--E 6
+
+--S 7
+(x1,y1):=(-3,-8)
+--R
+--R (7) - 8
+--R Type:
Integer
+--E 7
+
+--S 8
+(x2,y2):=(-6,2)
+--R
+--R (8) 2
+--R Type:
PositiveInteger
+--E 8
+
+--S 9
+m:=(y2-y1)/(x2-x1)
+--R
+--R 10
+--R (9) - --
+--R 3
+--R Type: Fraction
Integer
+--E 9
+
+--S 10
+solve(y1=m*x1+b,b)
+--R
+--R (10) [b= - 18]
+--R Type: List Equation Fraction Polynomial
Integer
+--E 10
+
+--S 11
+b:=-18
+--R
+--R (11) - 18
+--R Type:
Integer
+--E 11
+
+--S 12
+y=m*x+b
+--R
+--R 10
+--R (12) y= - -- x - 18
+--R 3
+--R Type: Equation Polynomial Fraction
Integer
+--E 12
+)spool
+)lisp (bye)
+
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}
+
+
diff --git a/src/input/classtalk.input.pamphlet
b/src/input/classtalk.input.pamphlet
new file mode 100644
index 0000000..7f89df1
--- /dev/null
+++ b/src/input/classtalk.input.pamphlet
@@ -0,0 +1,733 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input classtalk.input}
+\author{Timothy Daly}
+\maketitle
+\begin{abstract}
+These are examples from the talk ``Axiom in an Educational Setting''.
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+<<*>>=
+)spool classtalk.output
+)set message test on
+)set message auto off
+)set break resume
+)clear all
+
+@
+\section{Numbers}
+<<*>>=
+--S 1
+1
+--R
+--R (1) 1
+--R Type:
PositiveInteger
+--E 1
+
+--S 2
+1/2
+--R
+--R 1
+--R (2) -
+--R 2
+--R Type: Fraction
Integer
+--E 2
+
+--S 3
+3+4*%i
+--R
+--R (3) 3 + 4%i
+--R Type: Complex
Integer
+--E 3
+
+--S 4
+3.4
+--R
+--R (4) 3.4
+--R Type:
Float
+--E 4
+
+--S 5
+X::ROMAN
+--R
+--R (5) X
+--R Type:
RomanNumeral
+--E 5
+
+--S 6
+binary(5)
+--R
+--R (6) 101
+--R Type:
BinaryExpansion
+--E 6
+
+--S 7
+factor(60)
+--R
+--R 2
+--R (7) 2 3 5
+--R Type: Factored
Integer
+--E 7
+
+--S 8
+q:=(y-1)*x*(z+5)
+--R
+--R (8) (x y - x)z + 5x y - 5x
+--R Type: Polynomial
Integer
+--E 8
+
+--S 9
+factor q
+--R
+--R (9) x(y - 1)(z + 5)
+--R Type: Factored Polynomial
Integer
+--E 9
+
+--S 10
+eval(q,[x=5,y=6,z=7])
+--R
+--R (10) 300
+--R Type: Polynomial
Integer
+--E 10
+
+--S 11
+eval(q,[x=5,y=6])
+--R
+--R (11) 25z + 125
+--R Type: Polynomial
Integer
+--E 11
+
+@
+\section{Trigonometry}
+<<*>>=
+--S 12
+b:=[log a, exp a, asin a, acos a, atan a, acot a, sinh a]
+--R
+--R a
+--R (12) [log(a),%e ,asin(a),acos(a),atan(a),acot(a),sinh(a)]
+--R Type: List Expression
Integer
+--E 12
+
+--S 13
+[exp b.1, log b.2, sin b.3, cos b.4, tan b.5, cot b.6, asinh b.7]
+--R
+--R (13) [a,a,a,a,a,a,a]
+--R Type: List Expression
Integer
+--E 13
+
+--S 14
+a:=.7
+--R
+--R (14) 0.7
+--R Type:
Float
+--E 14
+
+--S 15
+b:=[log a, exp a, asin a, acos a, atan a, acot a, sinh a]
+--R
+--R (15)
+--R [- 0.3566749439 3873237891, 2.0137527074 704765216, 0.7753974966
1075306374,
+--R 0.7953988301 8414355549, 0.6107259643 8920861654, 0.9600703624
0568800269,
+--R 0.7585837018 3953350346]
+--R Type: List
Float
+--E 15
+
+--S 16
+[exp b.1, log b.2, sin b.3, cos b.4, tan b.5, cot b.6, asinh b.7]
+--R
+--R (16) [0.7,0.7,0.7,0.7,0.7,0.7,0.7]
+--R Type: List
Float
+--E 16
+
+--S 17
+simplify(sin(x)**2+cos(x)**2)
+--R
+--R (17) 1
+--R Type: Expression
Integer
+--E 17
+
+@
+\section{Polynomial Manipulations}
+<<*>>=
+)clear all
+--S 18
+eq1:=A*x^2 + B*x*y + C*y^2 + D*x + E*y + F
+--R
+--R 2 2
+--R (1) C y + (B x + E)y + A x + D x + F
+--R Type: Polynomial
Integer
+--E 18
+
+--S 19
+rotatex:=x'*cos(t)-y'*sin(t)
+--R
+--R (2) - y' sin(t) + x' cos(t)
+--R Type: Expression
Integer
+--E 19
+
+--S 20
+rotatey:=x'*sin(t)+y'*cos(t)
+--R
+--R (3) x' sin(t) + y' cos(t)
+--R Type: Expression
Integer
+--E 20
+
+--S 21
+eval(eq1,[x=rotatex, y=rotatey])
+--R
+--R (4)
+--R 2 2 2
+--R (A y' - B x' y' + C x' )sin(t)
+--R +
+--R 2 2
+--R ((- B y' + (2C - 2A)x' y' + B x' )cos(t) - D y' + E x')sin(t)
+--R +
+--R 2 2 2
+--R (C y' + B x' y' + A x' )cos(t) + (E y' + D x')cos(t) + F
+--R Type: Expression
Integer
+--E 21
+
+@
+\section{Polynomials over Simple Algebraic Extension Fields}
+<<*>>=
+)clear all
+--S 22
+a:=rootOf(a^2+a+1)
+--R
+--R (1) a
+--R Type:
AlgebraicNumber
+--E 22
+
+--S 23
+factor(x^2+3)
+--R
+--R 2
+--R (2) x + 3
+--R Type: Factored Polynomial
Integer
+--E 23
+
+--S 24
+factor(x^2+3,[a])
+--R
+--R (3) (x - 2a - 1)(x + 2a + 1)
+--R Type: Factored Polynomial
AlgebraicNumber
+--E 24
+
+--S 25
+definingPolynomial(a)
+--R
+--R 2
+--R (4) a + a + 1
+--R Type:
AlgebraicNumber
+--E 25
+
+--S 26
+zerosOf(b^2+b+1,b)
+--R
+--R +---+ +---+
+--R \|- 3 - 1 - \|- 3 - 1
+--R (5) [----------,------------]
+--R 2 2
+--R Type: List Expression
Integer
+--E 26
+
+@
+\section{Derivatives}
+<<*>>=
+--S 27
+differentiate(sin(x),x)
+--R
+--R (6) cos(x)
+--R Type: Expression
Integer
+--E 27
+
+--S 28
+differentiate(sin(x),x,2)
+--R
+--R (7) - sin(x)
+--R Type: Expression
Integer
+--E 28
+
+--S 29
+differentiate(cos(z)/(x^2+y^3),[x,y,z],[1,2,3])
+--R
+--R 4 3
+--R (- 84x y + 24x y)sin(z)
+--R (8) --------------------------------
+--R 12 2 9 4 6 6 3 8
+--R y + 4x y + 6x y + 4x y + x
+--R Type: Expression
Integer
+--E 29
+
+--S 30
+y:=operator y
+--R
+--R (9) y
+--R Type:
BasicOperator
+--E 30
+
+--S 31
+deqx:=D(y(x),x,2)+D(y(x),x)+y(x)
+--R
+--R
+--R ,, ,
+--R (10) y (x) + y (x) + y(x)
+--R
+--R Type: Expression
Integer
+--E 31
+
+--S 32
+solve(deqx,y,x)
+--R
+--R x x
+--R +-+ - - - - +-+
+--R x\|3 2 2 x\|3
+--R (11) [particular= 0,basis= [cos(-----)%e ,%e sin(-----)]]
+--R 2 2
+--RType: Union(Record(particular: Expression Integer,basis: List Expression
Integer),...)
+--E 32
+
+@
+\section{Limits}
+<<*>>=
+)clear all
+--S 33
+limit((x^2-3*x+2)/(x^2-1),x=1)
+--R
+--R 1
+--R (1) - -
+--R 2
+--R Type: Union(OrderedCompletion Fraction Polynomial
Integer,...)
+--E 33
+
+--S 34
+limit(x*log(x),x=0)
+--R
+--R (2) [leftHandLimit= "failed",rightHandLimit= 0]
+--RType: Union(Record(leftHandLimit: Union(OrderedCompletion Expression
Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression
Integer,"failed")),...)
+--E 34
+
+--S 35
+limit(sinh(a*x)/tan(b*x),x=0)
+--R
+--R a
+--R (3) -
+--R b
+--R Type: Union(OrderedCompletion Expression
Integer,...)
+--E 35
+
+--S 36
+limit(sqrt(3*x^2+1)/(5*x),x=%plusInfinity)
+--R
+--R +-+
+--R \|3
+--R (4) ----
+--R 5
+--R Type: Union(OrderedCompletion Expression
Integer,...)
+--E 36
+
+--S 37
+complexLimit((2+z)/(1-z),z=%infinity)
+--R
+--R (5) - 1
+--R Type: OnePointCompletion Fraction Polynomial
Integer
+--E 37
+
+@
+\section{Indefinite Integration}
+<<*>>=
+)clear all
+--S 38
+integrate(1+sqrt(x)/x,x)
+--R
+--R +-+
+--R (1) 2\|x + x
+--R Type: Union(Expression
Integer,...)
+--E 38
+
+--S 39
+integrate(sin(x)/x,x)
+--R
+--R (2) Si(x)
+--R Type: Union(Expression
Integer,...)
+--E 39
+
+@
+This used to give the answer:
+$$\frac{\sqrt{x}\sqrt{\pi} erf(x\sqrt{a})}{2a}$$
+<<*>>=
+--S 40
+integrate(exp(-a*x^2),x)
+--R
+--R x 2
+--R ++ - %Q a
+--R (3) | %e d%Q
+--R ++
+--R Type: Union(Expression
Integer,...)
+--E 40
+
+--S 41
+integrate(sin(x)/x^2,x)
+--R
+--R x
+--R ++ sin(%Q)
+--R (4) | ------- d%Q
+--R ++ 2
+--R %Q
+--R Type: Union(Expression
Integer,...)
+--E 41
+
+@
+\section{Definite Integration}
+<<*>>=
+)clear all
+--S 42
+integrate(exp(-x)/sqrt(x),x=0..%plusInfinity)
+--R
+--R _ 1
+--R (1) | (-)
+--R 2
+--R Type: Union(f1: OrderedCompletion Expression
Integer,...)
+--E 42
+
+--S 43
+integrate(1/x^2,x=-1..1)
+--R
+--R
+--RDaly Bug
+--R >> Error detected within library code:
+--R integrate: pole in path of integration
+--R
+--R Continuing to read the file...
+--R
+--E 43
+
+)clear all
+
+@
+This used to return
+$$\frac{4\log{(4)}-8\log{(2)}+3\pi}{12}$$
+<<*>>=
+--S 44
+integrate(sin(x)^3/(sin(x)^3+cos(x)^3),x=0..%pi/2,"noPole")
+--R
+--R 2log(16) - 4log(4) + 3%pi
+--R (1) -------------------------
+--R 12
+--R Type: Union(f1: OrderedCompletion Expression
Integer,...)
+--E 44
+
+--S 45
+integrate(exp(-x^2)*log(x)^2,x=0..%plusInfinity)
+--R
+--R _ 1 1 _ 1 1 2
+--R | (-)polygamma(1,-) + | (-)digamma(-)
+--R 2 2 2 2
+--R (2) --------------------------------------
+--R 8
+--R Type: Union(f1: OrderedCompletion Expression
Integer,...)
+--E 45
+
+@
+\section{Laplace Transformations}
+<<*>>=
+)clear all
+
+--S 46
+laplace(sin(a*t)*cosh(a*t)-cos(a*t)*sinh(a*t),t,s)
+--R
+--R 3
+--R 4a
+--R (1) --------
+--R 4 4
+--R s + 4a
+--R Type: Expression
Integer
+--E 46
+
+--S 47
+laplace(2/t * (1-cos(a*t)),t,s)
+--R
+--R 2 2
+--R (2) log(s + a ) - 2log(s)
+--R Type: Expression
Integer
+--E 47
+
+--S 48
+laplace((exp(a*t)-exp(b*t))/t,t,s)
+--R
+--R (3) - log(s - a) + log(s - b)
+--R Type: Expression
Integer
+--E 48
+
+--S 49
+laplace(exp(a*t+b)*Ei(c*t),t,s)
+--R
+--R b s + c - a
+--R %e log(---------)
+--R c
+--R (4) -----------------
+--R s - a
+--R Type: Expression
Integer
+--E 49
+
+@
+\section{Clifford Algebras}
+{\tt CliffordAlgebra(n,K,Q)} defines a vector space of dimension $2^n$
+over $K$, given a quadratic form $Q$ on $K^n$ (e.q. quaternions).
+<<*>>=
+)clear all
+--S 50
+K:=Fraction Polynomial Integer
+--R
+--R (1) Fraction Polynomial Integer
+--R Type:
Domain
+--E 50
+
+--S 51
+qf:QFORM(2,K):=quadraticForm matrix([[-1,0],[0,-1]])$(SQMATRIX(2,K))
+--R
+--R +- 1 0 +
+--R (2) | |
+--R + 0 - 1+
+--R Type: QuadraticForm(2,Fraction Polynomial
Integer)
+--E 51
+
+--S 52
+i:=e(1)$CLIF(2,K,qf)
+--R
+--R (3) e
+--R 1
+--R Type: CliffordAlgebra(2,Fraction Polynomial
Integer,MATRIX)
+--E 52
+
+--S 53
+j:=e(2)$CLIF(2,K,qf)
+--R
+--R (4) e
+--R 2
+--R Type: CliffordAlgebra(2,Fraction Polynomial
Integer,MATRIX)
+--E 53
+
+--S 54
+k:=i*j
+--R
+--R (5) e e
+--R 1 2
+--R Type: CliffordAlgebra(2,Fraction Polynomial
Integer,MATRIX)
+--E 54
+
+--S 55
+x:=a+b*i+c*j+d*k
+--R
+--R (6) a + b e + c e + d e e
+--R 1 2 1 2
+--R Type: CliffordAlgebra(2,Fraction Polynomial
Integer,MATRIX)
+--E 55
+
+--S 56
+y:=m+f*i+g*j+h*k
+--R
+--R (7) m + f e + g e + h e e
+--R 1 2 1 2
+--R Type: CliffordAlgebra(2,Fraction Polynomial
Integer,MATRIX)
+--E 56
+
+--S 57
+x+y
+--R
+--R (8) m + a + (f + b)e + (g + c)e + (h + d)e e
+--R 1 2 1 2
+--R Type: CliffordAlgebra(2,Fraction Polynomial
Integer,MATRIX)
+--E 57
+
+--S 58
+x*y
+--R
+--R (9)
+--R a m - d h - c g - b f + (b m + c h - d g + a f)e
+--R 1
+--R +
+--R (c m - b h + a g + d f)e + (d m + a h + b g - c f)e e
+--R 2 1 2
+--R Type: CliffordAlgebra(2,Fraction Polynomial
Integer,MATRIX)
+--E 58
+
+@
+\section{Taylor Series}
+<<*>>=
+)clear all
+--S 59
+taylor(sin(x),x=0)
+--R
+--R 1 3 1 5 1 7 1 9 11
+--R (1) x - - x + --- x - ---- x + ------ x + O(x )
+--R 6 120 5040 362880
+--R Type: UnivariateTaylorSeries(Expression
Integer,x,0)
+--E 59
+
+@
+\section{Laurent Series}
+<<*>>=
+--S 60
+laurent(x/log(x),x=1)
+--R
+--R (2)
+--R - 1 3 5 1 2 11 3 11
4
+--R (x - 1) + - + -- (x - 1) - -- (x - 1) + --- (x - 1) - ---- (x - 1)
+--R 2 12 24 720 1440
+--R +
+--R 271 5 13 6 7297 7 425 8
+--R ----- (x - 1) - ---- (x - 1) + ------- (x - 1) - ------ (x - 1)
+--R 60480 4480 3628800 290304
+--R +
+--R 530113 9 10
+--R --------- (x - 1) + O((x - 1) )
+--R 479001600
+--R Type: UnivariateLaurentSeries(Expression
Integer,x,1)
+--E 60
+
+@
+\section{Puiseux Series}
+<<*>>=
+--S 61
+puiseux(sqrt(sec(x)),x=3*%pi/2)
+--R
+--R
+--R 1 3 7
+--R - - - -
+--R 3%pi 2 1 3%pi 2 1 3%pi 2 3%pi 5
+--R (3) (x - ----) + -- (x - ----) + --- (x - ----) + O((x - ----) )
+--R 2 12 2 160 2 2
+--R Type: UnivariatePuiseuxSeries(Expression
Integer,x,(3*pi)/2)
+--E 61
+
+@
+\section{General Series}
+<<*>>=
+--S 62
+series(x^x,x=0)
+--R
+--R (4)
+--R 2 3 4 5
+--R log(x) 2 log(x) 3 log(x) 4 log(x) 5
+--R 1 + log(x)x + ------- x + ------- x + ------- x + ------- x
+--R 2 6 24 120
+--R +
+--R 6 7 8 9 10
+--R log(x) 6 log(x) 7 log(x) 8 log(x) 9 log(x) 10
11
+--R ------- x + ------- x + ------- x + ------- x + -------- x + O(x
)
+--R 720 5040 40320 362880 3628800
+--R Type: GeneralUnivariatePowerSeries(Expression
Integer,x,0)
+--E 62
+
+@
+\section{Matrices}
+<<*>>=
+)clear all
+--S 63
+m:=matrix [[1,2],[3,4]]
+--R
+--R +1 2+
+--R (1) | |
+--R +3 4+
+--R Type: Matrix
Integer
+--E 63
+
+--S 64
+4*m*(-5)
+--R
+--R +- 20 - 40+
+--R (2) | |
+--R +- 60 - 80+
+--R Type: Matrix
Integer
+--E 64
+
+--S 65
+n:=matrix [[1,0,-2],[-3,5,1]]
+--R
+--R + 1 0 - 2+
+--R (3) | |
+--R +- 3 5 1 +
+--R Type: Matrix
Integer
+--E 65
+
+--S 66
+m*n
+--R
+--R +- 5 10 0 +
+--R (4) | |
+--R +- 9 20 - 2+
+--R Type: Matrix
Integer
+--E 66
+
+--S 67
+hilb:=matrix([[1/(i+j) for i in 1..3] for j in 1..3])
+--R
+--R +1 1 1+
+--R |- - -|
+--R |2 3 4|
+--R | |
+--R |1 1 1|
+--R (5) |- - -|
+--R |3 4 5|
+--R | |
+--R |1 1 1|
+--R |- - -|
+--R +4 5 6+
+--R Type: Matrix Fraction
Integer
+--E 67
+
+--S 68
+inverse(hilb)
+--R
+--R + 72 - 240 180 +
+--R | |
+--R (6) |- 240 900 - 720|
+--R | |
+--R + 180 - 720 600 +
+--R Type: Union(Matrix Fraction
Integer,...)
+--E 68
+
+@
+\section{Systems of Equations}
+<<*>>=
+)clear all
+--S 69
+solve([x+y+z=8,3*x-2*y+z=0,x+2*y+2*z=17],[x,y,z])
+--R
+--R (1) [[x= - 1,y= 2,z= 7]]
+--R Type: List List Equation Fraction Polynomial
Integer
+--E 69
+
+--S 70
+solve([x+2*y+3*z=2,2*x+3*y+4*z=2,3*x+4*y+5*z=2],[x,y,z])
+--R
+--R (2) [[x= %W - 2,y= - 2%W + 2,z= %W]]
+--R Type: List List Equation Fraction Polynomial
Integer
+--E 70
+
+--S 71
+solve([[1,1,1],[3,-2,1],[1,2,2]],[8,0,17])
+--R
+--R (3) [particular= [- 1,2,7],basis= [[0,0,0]]]
+--RType: Record(particular: Union(Vector Fraction Integer,"failed"),basis:
List Vector Fraction Integer)
+--E 71
+
+--S 72
+solve([[1,2,3],[2,3,4],[3,4,5]],[2,2,2])
+--R
+--R (4) [particular= [- 2,2,0],basis= [[1,- 2,1]]]
+--RType: Record(particular: Union(Vector Fraction Integer,"failed"),basis:
List Vector Fraction Integer)
+--E 72
+)spool
+)lisp (bye)
+
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}
diff --git a/src/input/intbypart.input.pamphlet
b/src/input/intbypart.input.pamphlet
index 72a04e5..0b26c61 100644
--- a/src/input/intbypart.input.pamphlet
+++ b/src/input/intbypart.input.pamphlet
@@ -91,6 +91,7 @@ integrate(x*exp(x),x)
--R Type: Union(Expression
Integer,...)
--E 2
@
+
\section{integrate $e^x cos(x) dx$}
This integral will require the substituion by parts rule to be applied twice.
@@ -165,8 +166,8 @@ integrate(x^3*exp(x^2),x)
--R 2
--R Type: Union(Expression
Integer,...)
--E 4
-@
+@
\section{integrate $ln(x^2+2)dx$}
To integrate
$$\int{ln(x^2+2)dx}$$
@@ -194,49 +195,123 @@ integrate(log(x^2+2),x)
--E 5
@
-\section{integrate $ln(x) dx$}
+\section{integrate $x\ sin(x) dx$}
To integrate
-$$\int{ln(x) dx}$$
+$$\int{x\ sin(x)\ dx}$$
Let
-$$u=ln(x)$$
-$$dv=dx$$
-$$du=\frac{1}{x}dx$$
-$$v=x$$
+$$u=x$$
+$$dv=sin(x)\ dx$$
+$$du = dx$$
+$$v = -cos(x)$$
so
-$$\int{ln(x)\ dx}$$
-$$=x\ ln(x)=\int{1\ dx}$$
-$$=x\ ln(x) - x + C$$
-$$=x(ln(x)-1)+C$$
+$$\int{x\ sin(x)\ dx}$$
+$$= -x\ cos(x) - \int{-cos(x)\ dx}$$
+$$= -x\ cos(x)+sin(x)+C$$
<<*>>=
--S 6
-integrate(log(x),x)
+integrate(x*sin(x),x)
--R
---R (6) x log(x) - x
+--R (6) sin(x) - x cos(x)
--R Type: Union(Expression
Integer,...)
--E 6
@
-\section{integrate $x\ sin(x) dx$}
+\section{integrate $x\ cos(x) dx$}
To integrate
-$$\int{x\ sin(x)\ dx}$$
+$$\int{x\ cos(x)\ dx}$$
Let
$$u=x$$
-$$dv=sin(x)\ dx$$
+$$dv=cos(x)\ dx$$
$$du = dx$$
-$$v = -cos(x)$$
+$$v = sin(x)$$
so
-$$\int{x\ sin(x)\ dx}$$
-$$= -x\ cos(x) - \int{-cos(x)\ dx}$$
-$$= -x\ cos(x)+sin(x)+C$$
+$$\int{x\ cos(x)\ dx}$$
+$$= x\ sin(x) - \int{sin(x)\ dx}$$
+$$= x\ sin(x)+cos(x)+C$$
<<*>>=
--S 7
-integrate(x*sin(x),x)
+integrate(x*cos(x),x)
+--R
--R
---R (7) sin(x) - x cos(x)
+--R (7) x sin(x) + cos(x)
--R Type: Union(Expression
Integer,...)
--E 7
@
+\section{integrate $x^2\ sin(x) dx$}
+To integrate
+$$\int{x^2\ sin(x)\ dx}$$
+Let
+$$u=x^2$$
+$$dv=sin(x)\ dx$$
+$$du = 2x\ dx$$
+$$v = -cos(x)$$
+so
+$$\int{x^2\ sin(x)\ dx}$$
+$$= -x^2\ cos(x) - \int{-2x\ cos(x)\ dx}$$
+$$= -x^2\ cos(x)+2\int{x\ cos(x)\ dx}$$
+$$=-x^2\ cos(x)+2(x\ sin(x)+cos(x))+C$$
+<<*>>=
+--S 8
+integrate(x^2*cos(x),x)
+--R
+--R
+--R 2
+--R (8) (x - 2)sin(x) + 2x cos(x)
+--R Type: Union(Expression
Integer,...)
+--E 8
+@
+
+\section{integrate $sin(x)\ cos(x)\ dx$}
+To integrate
+$$\int{sin(x)\ cos(x)\ dx}$$
+Let
+$$u=sin(x)$$
+$$dv=cos(x)\ dx$$
+$$du=cos(x)\ dx$$
+$$v=sin(x)$$
+so
+$$\int{sin(x)cos(x)dx}
+=sin(x)sin(x) - \int{sin(x)cos(x)dx}$$
+but the integral appears on both sides of the equation so
+$$2\int{sin(x)cos(x)dx}=sin^2(x)$$
+so
+$$\int{sin(x)cos(x)dx}=\frac{1}{2}sin^2(x)+C$$
+<<*>>=
+--S 9
+integrate(sin(x)*cos(x),x)
+--R
+--R
+--R 2
+--R cos(x)
+--R (9) - -------
+--R 2
+--R Type: Union(Expression
Integer,...)
+--E 9
+@
+
+\section{integrate $ln(x) dx$}
+To integrate
+$$\int{ln(x) dx}$$
+Let
+$$u=ln(x)$$
+$$dv=dx$$
+$$du=\frac{1}{x}dx$$
+$$v=x$$
+so
+$$\int{ln(x)\ dx}$$
+$$=x\ ln(x)=\int{1\ dx}$$
+$$=x\ ln(x) - x + C$$
+$$=x(ln(x)-1)+C$$
+<<*>>=
+--S 10
+integrate(log(x),x)
+--R
+--R (10) x log(x) - x
+--R Type: Union(Expression
Integer,...)
+--E 10
+@
+
\section{integrate $x^2\ ln(x)\ dx$}
To integrate
$$\int{x^2\ ln(x)\ dx}$$
@@ -251,15 +326,38 @@ $$\frac{x^3}{3} ln(x) - \int{\frac{x^3}{3}\frac{dx}{x}}$$
$$\frac{x^3}{3} ln(x)-\frac{1}{3}\int{x^2\ dx}$$
$$\frac{x^3}{3}ln(x)-\frac{1}{9}x^3 + C$$
<<*>>=
---S 8
+--S 11
integrate(x^2*log(x),x)
--R
---R 3 3
---R 3x log(x) - x
---R (8) --------------
---R 9
+--R 3 3
+--R 3x log(x) - x
+--R (11) --------------
+--R 9
--R Type: Union(Expression
Integer,...)
---E 8
+--E 11
+@
+
+\section{integrate $x^2\ e^x\ dx$}
+To integrate
+$$\int{x^2\ e^x\ dx}$$
+Let
+$$u=x^2$$
+$$dv=e^x\ dx$$
+$$du = 2x\ dx$$
+$$v = e^x$$
+so
+$$\int{x^2\ e^x\ dx}$$
+$$x^2\ e^x - 2x\ e^x - \int{e^x\ 2dx}$$
+$$x^2\ e^x - 2x\ e^x+2\ e^x+C$$
+<<*>>=
+--S 12
+integrate(x^2*exp(x),x)
+--R
+--R
+--R 2 x
+--R (12) (x - 2x + 2)%e
+--R Type: Union(Expression
Integer,...)
+--E 12
@
\section{integrate $sin^{-1}(x)\ dx$}
@@ -278,15 +376,128 @@ $$=x\ sin^{-1}(x) + \frac{1}{2}(2(1-x^2)^{1/2})+C$$
$$=x\ sin^{-1}(x)+(1-x^2)^{1/2}+C$$
$$=x\ sin^{-1}(x)+\sqrt{1-x^2}+C$$
<<*>>=
---S 9
-integrate(1/sin(x),x)
+--S 13
+integrate(asin(x),x)
+--R
--R
---R sin(x)
---R (9) log(----------)
---R cos(x) + 1
+--R +--------+
+--R | 2 +--------+
+--R 2x\|- x + 1 | 2
+--R - x atan(-------------) + 2\|- x + 1
+--R 2
+--R 2x - 1
+--R (13) --------------------------------------
+--R 2
--R Type: Union(Expression
Integer,...)
---E 9
-)spool
+--E 13
+@
+
+\section{integrate $\tan^{-1}(x)\ dx$}
+$$\int{\tan^{-1}(x)\ dx}$$
+Let
+$$u=tan^{-1}$$
+$$dv=dx$$
+$$du=\frac{1}{1+x^2}\ dx$$
+$$v=x$$
+so
+$$\int{\tan^{-1}(x)\ dx}$$
+$$=x\ \tan^{-1}-\int{\frac{x}{1+x^2}\ dx}$$
+$$=x\ tan^{-1}(x)-\frac{1}{2}\int{\frac{2x}{1+x^2}\ dx}$$
+$$=x\ tan^{-1}(x)-\frac{1}{2}ln(1+x^2)+C$$
+<<*>>=
+--S 14
+integrate(atan(x),x)
+--R
+--R
+--R 2 2x
+--R - log(x + 1) - x atan(------)
+--R 2
+--R x - 1
+--R (14) ------------------------------
+--R 2
+--R Type: Union(Expression
Integer,...)
+--E 14
+@
+
+\section{integrate $\sec^3(x)\ dx$}
+$$\int{\sec^3(x)\ dx}$$
+Let
+$$u=sec(x)$$
+$$dv=sec^2(x)\ dx$$
+$$du=sec(x)tan(x)\ dx$$
+$$v=tan(x)$$
+so
+$$\int{\sec^3(x)\ dx}$$
+$$=sec(x)tan(x)-\int{sec(x)tan^2(x)\ dx}$$
+$$=sec(x)tan(x)-\int{sec(x)(sec^2(x)-1)\ dx}$$
+$$=sec(x)tan(x)-\int{sec^3(x)\ dx}+\int{sec(x)\ dx}$$
+$$=sec(x)tan(x)-\int{sec^3(x)\ dx}+ln(\vert sec(x)+\tan(x)\vert )$$
+but
+$$=2\int{sec^3(x)\ dx}=sec(x)tan(x)+ln(\vert sec(x)+\tan(x)\vert )$$
+so
+$$\int{sec^3(x)\ dx}=
+\frac{1}{2}(sec(x)tan(x)+ln(\vert sec(x)+\tan(x)\vert ))+C$$
+<<*>>=
+--S 15
+integrate(sec(x)^3,x)
+--R
+--R
+--R (15)
+--R 2 sin(x) + cos(x) + 1 2 sin(x) - cos(x) - 1
+--R cos(x) log(-------------------) - cos(x) log(-------------------) +
sin(x)
+--R cos(x) + 1 cos(x) + 1
+--R
--------------------------------------------------------------------------
+--R 2
+--R 2cos(x)
+--R Type: Union(Expression
Integer,...)
+--E 15
+@
+
+\section{integrate $x^3\ e^{2x}$}
+$$\int{x^3\ e^{2x}\ dx}$$
+Let
+$$u=x^3$$
+$$dv=e^{2x}\ dx$$
+$$du=3x^2\ dx$$
+$$v=\frac{1}{2}e^{2x}$$
+$$\int{x^3\ e^{2x}\ dx}$$
+$$=\frac{1}{2}x^3\ e^{2x} - \frac{3}{2}\int{x^2\ e^{2x}\ dx}$$
+
+To solve
+$$\frac{3}{2}\int{x^2\ e^{2x}\ dx}$$
+Let
+$$u=x^2$$
+$$dv=e^{2x}\ dx$$
+$$du=2x\ dx$$
+$$v=\frac{1}{2}e^{2x}$$
+so after substitution the new result is
+$$=\frac{1}{2}x^3\ e^{2x} -
+\frac{3}{2}\left(\frac{1}{2}x^2\ e^{2x}-\int{xe^{2x}\ dx}\right)$$
+$$=\frac{1}{2}x^3\ e^{2x} - \frac{3}{4}x^2e^{2x}+
+\frac{3}{2}\int{xe^{2x}\ dx}$$
+Let
+$$u=x$$
+$$dv=e^{2x}\ dx$$
+$$du=dx$$
+$$v=\frac{1}{2}e^{2x}$$
+so
+$$\int{x^3\ e^{2x}\ dx}$$
+$$=\frac{1}{2}x^3\ e^{2x}-\frac{3}{4}x^2\ e^{2x}+
+\frac{3}{2}\left(\frac{1}{2}xe^{2x}-\frac{1}{2}\int{e^{2x}\ dx}\right)$$
+$$=\frac{1}{2}x^3\ e^{2x}-\frac{3}{4}x^2\ e^{2x}+
+\frac{3}{4}xe^{2x}-\frac{3}{8}e^{2x}+C$$
+<<*>>=
+--S 16
+integrate(x^3*exp(2*x),x)
+--R
+--R
+--R 3 2 2x
+--R (4x - 6x + 6x - 3)%e
+--R (16) ------------------------
+--R 8
+--R Type: Union(Expression
Integer,...)
+--E 16
+)spool
)lisp (bye)
@
diff --git a/src/input/limit.input.pamphlet b/src/input/limit.input.pamphlet
index c210c87..ec80f68 100644
--- a/src/input/limit.input.pamphlet
+++ b/src/input/limit.input.pamphlet
@@ -5,24 +5,276 @@
\author{Timothy Daly}
\maketitle
\begin{abstract}
+Exercise the limit function.
\end{abstract}
\eject
\tableofcontents
\eject
-\section{License}
-<<license>>=
---Copyright The Numerical Algorithms Group Limited 1991.
-@
+\section{Limit of a Function}
+If $f$ is a function, then
+$$\lim_{x -> a}{f(x) = A}$$
+if the value of $f(x)$ gets arbitrarily close to $A$ as $x$ gets
+arbitrarily close to $a$. For example,
+$$\lim_{x->3}{x^2}=9$$
+since $x^2$ gets arbitrarily close to 9 as $x$ approaches 3.
+
+By definition, the limit
+$$\lim_{x -> a}{f(x) = A}$$ if and only if, for any chosen positive
+number $\epsilon$, however small, there exists a positive number
+$\delta$ such that
+$$0 < \vert x-a \vert < \delta {\tt\ implies\ }
+\vert f(x)-A \vert < \epsilon$$
+Note that $f(x)$ does not need to be defined at $a$.
+
<<*>>=
)spool limit.output
)set message test on
)set message auto off
)clear all
+--S 1 of 15
+limit((x^2-4)/(x-2),x=2)
+--R
+--R
+--R (1) 4
+--R Type: Union(OrderedCompletion Fraction Polynomial
Integer,...)
+--E 1
+@
+\section{Right and Left Limits}
+The limit
+$$\lim_{x -> a^-}{f(x) = A}$$
+given that $f$ is defined in the interval $(c,a)$ and $f(x)$
+approaches $A$ as $x$ approaches $a$ through the values less than
+$a$, that is, as $x$ approaches $a$ from the left.
+
+Similarly
+$$\lim_{x -> a^+}{f(x) = A}$$
+means that $f$ is defined in some interval $(a,d)$ and $f(x)$
+approaches $A$ as $x$ approaches $a$ from the right.
+
+If $f$ is defined in an interval to the left of $a$ and in an interval
+to the right of $a$ then the statement
+$$\lim_{x -> a}{f(x) = A}$$
+is equivalent to the conjunction of the two statements
+$$\lim_{x -> a^-}{f(x) = A} {\rm\ and\ } \lim_{x -> a^+}{f(x) = A}$$.
+
+The existence of the limit from the left does not imply the existence
+of the limit from the right. The existence of the limit from the
+right does not imply the existence of the limit from the left.
+
+When a function is defined only on one side of a point $a$, then
+$$\lim_{x -> a}{f(x)}$$
+is identical with the one-sided limit, if it exists. A two-sided
+limit might not exist because it is not defined outside a certain
+bound. For example, if
+$$f(x)=\sqrt{x}$$
+then $f$ is defined only at and to the right of 0. Hence,
+$$\lim_{x->0}{\sqrt{x}}=\lim_{x->0^+}{\sqrt{x}}=0$$
+But the limit
+$$\lim_{x->0^-}{\sqrt{x}}$$
+does not exist since $\sqrt{x}$ is not defined for real values of $x$.
+
+A limit might not exist because the function grows without bound
+as it approaches the limit point. For example, the function
+$\sqrt{\frac{1}{x}}$ is only defined when $x > 0$. So
+$$\lim_{x->0^+}{\sqrt{\frac{1}{x}}}$$
+does not exist since $\frac{1}{x}$ gets larger as $x -> 0$ from
+the right. So
+$$\lim_{x->0}{\sqrt{\frac{1}{x}}}$$
+does not exist.
+
+<<*>>=
+--S 2 of 15
+limit(sqrt(9-x^2),x=-4)
+--R
+--R
+--R (2) "failed"
+--R Type:
Union("failed",...)
+--E 2
+
+--S 3 of 15
+limit(sqrt(9-x^2),x=-3)
+--R
+--R
+--R (3) [leftHandLimit= "failed",rightHandLimit= 0]
+--RType: Union(Record(leftHandLimit: Union(OrderedCompletion Expression
Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression
Integer,"failed")),...)
+--E 3
+
+--S 4 of 15
+limit(sqrt(9-x^2),x=-2)
+--R
+--R
+--R +-+
+--R (4) \|5
+--R Type: Union(OrderedCompletion Expression
Integer,...)
+--E 4
+
+--S 5 of 15
+limit(sqrt(9-x^2),x=0)
+--R
+--R
+--R (5) 3
+--R Type: Union(OrderedCompletion Expression
Integer,...)
+--E 5
+
+--S 6 of 15
+limit(sqrt(9-x^2),x=2)
+--R
+--R
+--R +-+
+--R (6) \|5
+--R Type: Union(OrderedCompletion Expression
Integer,...)
+--E 6
+
+--S 7 of 15
+limit(sqrt(9-x^2),x=3)
+--R
+--R
+--R (7) [leftHandLimit= 0,rightHandLimit= "failed"]
+--RType: Union(Record(leftHandLimit: Union(OrderedCompletion Expression
Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression
Integer,"failed")),...)
+--E 7
+
+--S 8 of 15
+limit(sqrt(9-x^2),x=4)
+--R
+--R
+--R (8) "failed"
+--R Type:
Union("failed",...)
+--E 8
+
@
+\section{Theorems on Limits}
+If $$f(x)=c$$ where $c$ is a constant, then
+$$\lim_{x->a}{f(x)=c}$$
+
+Assume that
+$$\lim_{x->a}{f(x)=A}{\rm\ and\ }\lim_{x->a}{g(x)=B}$$
+then the limit of a constant times a function is
+the constant times limit of a function is
+the constant times the limit.
+$$\lim_{x->a}{cf(x)}=
+c*\lim_{x->a}{f(x)=
+cA}$$
+
+The limit of a function plus (or minus) another function is
+the limit of the first function plus (or minus)
+the limit of the second function is
+the sum (or difference) of their limits.
+$$\lim_{x->a}{[f(x)\pm{}g(x)]}=
+\lim_{x->a}{f(x)}\pm\lim_{x->a}{g(x)}=
+A\pm{}B$$
+
+The limit of the product of functions is
+the limit of the first function times the limit of the second function is
+the product of their limits.
+$$\lim_{x->a}{[f(x)g(x)]}=
+\lim_{x->a}{f(x)}*\lim_{x->a}{g(x)}=
+A*B$$
+
+The limit of the ratio of two functions is
+the the limit of the first function divided by the limit of the
+second function provided the limit of the second function is non-zero.
+$$\lim_{x->a}{\left(\frac{f(x)}{g(x)}\right)}=
+\frac{\lim_{x->a}{f(x)}}{\lim_{x->a}{g(x)}}=
+\frac{A}{B},
+{\rm\ if\ }B\ne{}0$$
+
+The limit of the ${\rm{}n}^{th}$ root of a function is
+the ${\rm{}n}^{th}$ root of the limit of the function provided
+the ${\rm{}n}^{th}$ root exists.
+$$\lim_{x->a}{\sqrt[n]{f(x)}}=\sqrt[n]{\lim_{x->a}{f(x)}}=
+\sqrt[n]{A},{\rm\ if\ }\sqrt[n]{A}{\rm\ is\ defined}$$
+
The returned limit seems to be wrong:
+
+\section{Infinity}
+Let
+$$\lim_{x->a}{f(x)}=+\infty$$
+mean that, as $x$ approaches $a$, $f(x)$ eventually becomes and
+therefore remains greater than any preassigned positive number,
+however large. In such a case, we say that $f(x)$ approaches
+$+\infty$ as $x$ approaches $a$. The
+$$\lim_{x->a}{f(x)}=+\infty$$
+if and only if, for any positive number $M$, there exists a
+positive number $\delta$ such that, whenever
+$$0 < \vert x-a \vert < \delta {\rm\ then\ }f(x) > M$$
+
+Similarly, let
+$$\lim_{x->a}{f(x)}=-\infty$$
+mean that, as $x$ approaches $a$, $f(x)$ eventually becomes and
+thereafter remains less than any preassigned negative number. In
+that case, we say that $f(x)$ approaches
+$-\infty$ as $x$ approaches $a$.
+
+Let
+$$\lim_{x->a}{f(x)}=\infty$$
+mean that, as $x$ approaches $a$, $\vert f(x)\vert$ eventually becomes and
+thereafter remains greater than any preassigned positive number. Hence,
+$$\lim_{x->a}{f(x)}=\infty$$ if and only if
+$$\lim_{x->a}{\vert f(x)\vert}=+\infty$$
+
+These definitions can be extended to one-sided limits.
+
+<<*>>=
+--S 9 of 15
+limit(1/x^2,x=0)
+--R
+--R
+--R (9) + infinity
+--R Type: Union(OrderedCompletion Fraction Polynomial
Integer,...)
+--E 9
+
+--S 10 of 15
+limit(-1/(x-1)^2,x=1)
+--R
+--R
+--R (10) - infinity
+--R Type: Union(OrderedCompletion Fraction Polynomial
Integer,...)
+--E 10
+
+--S 11 of 15
+limit(1/x,x=0)
+--R
+--R
+--R (11) [leftHandLimit= - infinity,rightHandLimit= + infinity]
+--RType: Union(Record(leftHandLimit: Union(OrderedCompletion Fraction
Polynomial Integer,"failed"),rightHandLimit: Union(OrderedCompletion Fraction
Polynomial Integer,"failed")),...)
+--E 11
+
+@
+The limit concepts already introduced can be extended to the case
+where the variable approaches $+\infty$ or $-\infty$. For example,
+$$\lim_{x->+\infty}{f(x)}=A$$
+means that $f(x)$ approaches $A$ as $x -> +\infty$ or, in more
+precise terms, given any positive $\epsilon$, there exists a
+number $N$ such that, whenever $x>N$, then
+$$\vert f(x)-A\vert < \epsilon$$
+
+Similar statements can be made for:
+$$\lim_{x->-\infty}{f(x)}=A$$
+$$\lim_{x->+\infty}{f(x)}=+\infty$$
+$$\lim_{x->-\infty}{f(x)}=-\infty$$
+$$\lim_{x->+\infty}{f(x)}=-\infty$$
+$$\lim_{x->-\infty}{f(x)}=+\infty$$
<<*>>=
---S 1 of 2
+--S 12 of 15
+limit(1/x,x=%plusInfinity)
+--R
+--R
+--R (12) 0
+--R Type: Union(OrderedCompletion Fraction Polynomial
Integer,...)
+--E 12
+
+--S 13 of 15
+limit(2+(1/x^2),x=%plusInfinity)
+--R
+--R
+--R (13) 2
+--R Type: Union(OrderedCompletion Fraction Polynomial
Integer,...)
+--E 13
+
+)clear all
+
+--S 14 of 15
f := exp(n) * (sin(1/n + exp(-n)) - sin(1/n))
--R
--R
@@ -31,22 +283,24 @@ f := exp(n) * (sin(1/n + exp(-n)) - sin(1/n))
--R (1) %e sin(-----------) - %e sin(-)
--R n n
--R Type: Expression
Integer
---E 1
+--E 14
---S 2 of 2
+--S 15 of 15
limit(f,n=%plusInfinity)
--R
--R
--R (2) "failed"
--R Type:
Union("failed",...)
---E 2
+--E 15
)spool
)lisp (bye)
@
\eject
\begin{thebibliography}{99}
-\bibitem{1} nothing
+\bibitem{1} Ayres, Frank Jr. and Mendelson, Elliott
+``Calculus'' Schaum's Outlines 4th edition 1999
+ISBN 0-07-041973-6 pp61-63
\end{thebibliography}
\end{document}
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